Multiple-description vector quantization with lattice codebooks: design and analysis

Abstract

The problem of designing a multiple-description vector quantizer with lattice codebook /spl Lambda/ is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A/sub 2/ and Z/sup i/, i=1, 2, 4, 8, that make use of this labeling algorithm. The high-rate squared-error distortions for this family of L-dimensional vector quantizers are then analyzed for a memoryless source with probability density function (PDF) p and differential entropy h(p)</spl infin/. For any a /spl epsiv/ (0, 1) and rate pair (R, R), it is shown that the two-channel distortion d~/sub o/ and the channel 1 (or channel 2) distortion d~/sub s/ satisfy lim/sub R/spl rarr//spl infin//d~/sub o/2/sup 2R(1+a)/= 1/4 G(/spl Lambda/)2/sup 2h(p)/ and lim/sub R/spl rarr//spl infin//d~/sub s/2/sup 2R(1-a)/=G(S/sub L/)2/sup 2h(p)/ where G(/spl Lambda/) is the normalized second moment of a Voronoi cell of the lattice /spl Lambda/ and G(S/sub L/) is the normalized second moment of a sphere in L dimensions.